Cross Product

Introduction

The cross product is an operation on vectors which is often used in physics. With scalars we know that if a quantity $c$ is proportional to two other quantities, $a$ and $b$, we define this relationship by the product $c=ab$. For vector quantities we would like to apply the same logic. So, if vector $\vec C$ is proportional to a vector $\vec A$ and a vector $\vec B$ we define $\vec C=\vec A \times \vec B$ as their cross product. However a vector is another quantity than a scalar and we need to define precisely the mechanics of this cross product. It must produce a vector $\vec C$ with a certain lenght which is representative for the magnitude of this product. This vector must have a well defined direction too.

It turns out that the magnitude of a product of two vectors is best described by the area of a parallelogram. We therefore first start to analyze the parallelogram.

Parallelogram

Let us start with a definition from Wikipedia.
Definition: Parallelogram
In Euclidean geometry, a parallelogram is a convex (angles less than or equal $180^\circ$) quadrilateral (four straight sides) with two pairs of parallel sides. The opposite or facing sides of a parallelogram are of equal length and the opposite angles of a parallelogram are of equal measure.

The congruence of opposite sides and opposite angles is a direct consequence of the Euclidean Parallel Postulate and neither condition can be proven without appealing to the Euclidean Parallel Postulate or one of its equivalent formulations.

There are different types of parallograms, a rhombus (four sides of equal length), a rectangle (four straight angles) and a square (four sides of equal lenght and four straight angles).

Area of a parallelogram

The area of a parallelogram is the area of a rectangle minus 2 times the area of a triangle (see the picture). The area of the rectangle: \[ \begin{align} A_r&=(A+T)H \end{align} \] The area of the triangle: \[ \begin{align} A_t&=\frac{1}{2}TH \end{align} \] The area of the parallelogram: \[ \begin{align} A_p&=(A+T)H-TH \\ &=AH\\ &=AB\sin \theta \end{align} \]

Observe that for a parallelogram only the rectangular side $H=B \sin \theta$ is taken in the product, which is of course smaller than the length of $B$.

We translate this formula to a formula with vectors involved:

The area of a parallelogram formed by two uectors $\vec u$ and $\vec v$ in $\mathbb{R}^n$ with an angle $\theta$ between them is defined by: \[ A= \norm{\mathbf{u}} \norm{\mathbf{v}} \sin \theta \]

We can derive a different form of this formula. Take the square of $O$: \[ \begin{align*} O^2&=\norm{\mathbf{u}}^2 \norm{\mathbf{v}}^2 \sin^2 \theta \\ &=\norm{\mathbf{u}}^2 \norm{\mathbf{v}}^2 (1- \cos^2) \theta \\ &=\norm{\mathbf{u}}^2 \norm{\mathbf{v}}^2 - \norm{\mathbf{u}}^2 \norm{\mathbf{v}}^2 \cos^2 \theta \\ &=(u,u)(v,v)-(u,v)^2 \tag{1} \end{align*} \]

With the above equation we can find a formula for the area of a parallelogram expressed in component form. Let us first illustrate in $\mathbb{R}^2$. Let $\vec u=(u_1,u_2)$ and $\vec v=(v_1,v_2)$ be two vectors. We try to find the area of the parallelogram with edges $\vec u,\vec v$, by algebraically working out the last formula. Note that the terms with the same subscripts cancel. \begin{align*} O^2&=(u_1^2+u_2^2)(v_1^2+v_2^2)-(u_1v_1+u_2v_2)^2 \\ &=u_1^2v_1^2+u_1^2v_2^2+u_2^2v_1^2+u_2^2v_2^2-u_1^2v_1^2-2u_1v_1u_2v_2-u_2^2v_2^2 \\ &=u_1^2v_2^2-2u_1v_1u_2v_2+u_2^2v_1^2 \\ &=\left(u_1v_2-u_2v_1\right)^2 \end{align*} So, \[ O=|u_1v_2-u_2v_1| \] This can also be expressed by an determinant: \[O=\mathrm{abs} \left| {\begin{array}{*{20}c} u_1 & u_2 \\ v_1 & v_2 \\ \end{array} } \right|=|u_1v_2-u_2v_1| \]

We can also take two vectors $\vec u=(u_1,u_2,u_3)$ and $\vec v=(v_1,v_2,v_3)$ from $\mathbb{R}^3$ and find the area of the parallelogram with edges $\vec u,\vec v$. We get the same systematic pattern, each square term with same subscripts cancel. \[ \begin{align} O^2&=\left(u_1v_2-u_2v_1\right)^2+\left(u_1v_3-u_3v_1\right)^2+\left(u_3v_2-u_2v_3\right)^2 \tag{2}\\ \end{align} \] The terms are the areas of the parallelograms of the projection of the parallogram with edges $\vec u$ and $\vec v$ in the three coordinate planes $xy$, $xz$ and $zy$.

Cross Product

We define the cross product of the vectors $\vec u$ and $\vec v$ , written $\vec u \times \vec v$ , to be a vector perpendicular to the plane defined by $\vec u$ and $\vec v$. The magnitude of this vector is the area of the parallelogram formed by the two vectors. The direction of $\vec u \times \vec v$ is given by the normal vector, $\vec n$ There are two choices for $\vec n$ , pointing out of the plane in opposite directions. We pick the direction given by the right-hand screw rule

Based on this definition and the formula of the area of a parellelogram we have following formula for the cross product.

Definition: Cross Product - Geometric
If $\vec u$ and $\vec v$ are vectors from $\mathbb{R}^3$ and non-zero and non-parallel vectors then the cross product is defined by: \[ \vec w = \vec u \times \vec v = (\text{area of parallelogram with edges } \vec u \text{ and } \vec v) \mathbf{\hat{n}}= \norm{\mathbf{u}} \norm{\mathbf{v}} \sin \theta \mathbf{\hat{n}} \] where $0 \leq \theta \leq \pi$ is the angle between $\vec u$ and $\vec v$ and $\mathbf{\hat{n}}$ is the unit vector perpendicular to $\vec u$ and $\vec v$ and in the direction given by the right-hand screw rule. If $\vec u$ and $\vec v$ are parallel then $\vec u \times \vec v =0$.

We now analyze the properties of this definition for the vector $\vec w$. The first property shows the algebraic consequence of the right-hand rule.

Proposition
Given a righ-handed orthonormal co-ordinate system, with unit vectors $\mathbf{\hat{e}}_x,\mathbf{\hat{e}}_y,\mathbf{\hat{e}}_z$, and three vectors $\vec u, \vec v, \vec w = \vec u \times \vec v$, then for the following determinant we have: \[ A=\left| {\begin{array}{*{20}c} u_1 & u_2 & u_3\\ v_1 & v_2 & v_3 \\ w_1 & w_2 & w_3 \\ \end{array} } \right|>0 \]
Proof
Now we analyze the algebraic consequence of the definition for the components of the cross product vector.
Proposition: Uniqueness and existence cross product

For every two non-zero, non-parallel vectors $\vec u$, $\vec v$ in $\mathbb{R}^3$, there is a unique vector $\vec w = \vec u \times \vec v$. If $(u_1, u_2, u_3)$ and $(v_1, v_2, v_3)$ are the components of $\vec u$ and $\vec v$ in a Cartesian right-handed system then the components $(w_1,w_2,w_3)$ of $\vec w$ are: \[ w_1 = u_2v_3 - u_3v_2, w_2 = u_3v_1 - u_1v_3, w_3 = u_1v_2 - u_2v_1 \tag{4} \] Note: $w_i=u_jv_k-u_kv_j$ with $i,j,k$ all different and a permutation of indices such that an even number of interchanges of adjacent elements has a positive sign and odd number a negative sign). This can symbolically be represented by: \[ \vec w = \vec u \times \vec v = \left| {\begin{array}{*{20}c} \mathbf{\hat{e}}_x & \mathbf{\hat{e}}_y & \mathbf{\hat{e}}_z \\ u_1 & u_2 & u_3\\ v_1 & v_2 & v_3 \\ \end{array} } \right| \tag{5} \]

Proof
Based on the rules of the determinant following properties of the cross product immediately follows.

Anti commutative

An odd number or inter-changes of rows (or columns) changes the sign of the determinant: \[ \vec u \times \vec v = - (\vec v \times \vec u) \]
The following properties follow form the linearity of the determinant in each row.

Scalar multiplication

Multiplying a row or a column by a factor multiplies the value of the determinant with the same factor: \[ \lambda\vec u \times \vec v = \vec u \times \lambda \vec v = \lambda (\vec u \times \vec v) \]

Distributive property

The determinant with row $u_i+v_i$ is equal to the sum of the determinants with this row replaced by $u_i$ and $v_i$: $\mathrm{det}(\ldots,u_i+v_i,\dots)=\mathrm{det}(\ldots,u_i,\dots)+\mathrm{det}(\ldots,v_i,\dots)$ \[ \vec r \times (\vec u + \vec v)= \vec r \times \vec u + \vec r \times \vec v \]
Based on what we learned from the above properties we can give an alternative property based definition of the cross product.
Definition: Cross Product - Algebraic

Let $n=3$, a cross product is defined which assigns to any two vectors $\vec v,\vec w \in \mathbb{R}^n$ a vector $\vec v \times \vec w \in \mathbb{R}^n$ such that the following four properties hold:

  • $\vec v \times \vec w$ is a bilinear function of $\vec v$ and $\vec w$
  • $(\vec v \times \vec w)\cdot \vec v=(\vec v \times \vec w)\cdot \vec w=\vec 0$, the cross product is perpendicular to $\vec v$ and $\vec w$
  • $\norm{\vec v \times \vec w}^2=\norm{\vec v}^2\norm{\vec w}^2-(\vec v \cdot \vec w)^2$, the magnitude of the cross product is the area of the parallelogram spanned by the vectors $\vec v, \vec w$
  • $\mathbf{\hat{e}_1}\times\mathbf{\hat{e}_2}=\mathbf{\hat{e}_3}$, where $\mathbf{\hat{e}_1},\mathbf{\hat{e}_2},\mathbf{\hat{e}_3}$ is an orthonormal right handed basis for $\mathbb{R}^n$

Coordinate systems

In all analysis we choose to work with a right handed system defined by: \begin{align*} \mathbf{\hat{e}_1} \times \mathbf{\hat{e}_2} &= \mathbf{\hat{e}_3} \\ \mathbf{\hat{e}_2} \times \mathbf{\hat{e}_3}&= \mathbf{\hat{e}_1} \\ \mathbf{\hat{e}_3} \times \mathbf{\hat{e}_1} &= \mathbf{\hat{e}_2} \\ \mathbf{\hat{e}_2} \times \mathbf{\hat{e}_1} &= -\mathbf{\hat{e}_3} \\ \mathbf{\hat{e}_1} \times \mathbf{\hat{e}_3} &= -\mathbf{\hat{e}_2} \\ \mathbf{\hat{e}_3} \times \mathbf{\hat{e}_3} &= -\mathbf{\hat{e}_1} \\ \end{align*} For a cyclic rotation of the basis vectors (i.e. $(\mathbf{\hat{e}_1},\mathbf{\hat{e}_2},\mathbf{\hat{e}_3}) \to (\mathbf{\hat{e}_2},\mathbf{\hat{e}_3},\mathbf{\hat{e}_1}) \to (\mathbf{\hat{e}_3},\mathbf{\hat{e}_1},\mathbf{\hat{e}_2})$) the sign is positive and for a acyclic rotation the sign is negative. A cyclic rotation corresponds to an even number of permutations (switches) and an acyclic rotation to an odd number of permutations.

In the following figure the even permutations or cyclic paths are shown in yellow (counter clockwise), the odd permuation or acyclic paths in red (clockwise) and the repeating indices in in blue.

Index notation

We have already seen that the Kronecker symbol allows us to write the component form of the dot product very compact as $\mathbf{\hat{e}_i} \cdot \mathbf{\hat{e}_j}=\delta_{ij}$. Also a symbol has been defined specifically for the cross product, the Levi-Civita symbol: \[ \varepsilon_{ijk} = \varepsilon^{ijk} = \begin{cases} +1 & \text{if } (i,j,k) \text{ is an even permutation of } (1,2,3), \\ -1 & \text{if } (i,j,k) \text{ is an odd permutation of } (1,2,3), \\ \;\;\,0 & (i,j,k) \text{ otherwise, repeating indices } \end{cases} \] We can apply this notation as follows: \[ (\mathbf{\hat{e}_i} \times \mathbf{\hat{e}_j})\cdot \mathbf{\hat{e}_k} = \varepsilon_{ijk} \] So we can write: \[ (\mathbf{\hat{e}_i} \times \mathbf{\hat{e}_j}) = \sum_{k=1}^3 \varepsilon_{ijk} \mathbf{\hat{e}_k} \] When the basis is orthogonal and right handed and $\vec u =\sum_{i=1}^3 u_i \mathbf{\hat{e}_i}$ and $\vec v =\sum_{j=1}^3 v_j \mathbf{\hat{e}_j}$ then: \begin{align*} \vec u \times \vec v &=\sum_{i=1}^3 \sum_{j=1}^3 u_iv_j (\mathbf{\hat{e}_i} \times \mathbf{\hat{e}_j}) \\ &=\sum_{i=1}^3 \sum_{j=1}^3 \sum_{k=1}^3 u_iv_j\varepsilon_{ijk}\mathbf{\hat{e}_k} \end{align*}

Now we see a summation symbol for and index $i,j,k$ where each index is repeated twice in the terms that are summed. It implies that we sum all possible combinations. The value of the indices can not be choosen freely. We call these type of indices dummy or bound indices opposed to free indices. Einstein noted that it is then redundant to include their summation sign. So we could rewrite this equation more compact as: \[ \vec u \times \vec v =\varepsilon_{ijk}u_iv_j\mathbf{\hat{e}_k} \]

The $l$ component of the cross product: \begin{align*} (\vec u \times \vec v) \cdot \mathbf{\hat{e}_l} &= u_iv_j\varepsilon_{ijk}\mathbf{\hat{e}_k} \cdot \mathbf{\hat{e}_l} =u_iv_j\varepsilon_{ijk}\delta_{kl} = u_iv_j\varepsilon_{ijl} = \varepsilon_{lmn}u_mv_n \end{align*} or switching $l$ to $i$ \[ \vec w = \vec u \times \vec v= \leftrightarrow w_i=\varepsilon_{ijk}u_jv_k \]

We can use these outcomes in working out the cross-product: \[ \begin{align*} \vec u \times \vec v&=(u_1\mathbf{\hat{e}}_1+u_2\mathbf{\hat{e}}_2+u_3\mathbf{\hat{e}}_3)(v_1\mathbf{\hat{e}}_1+v_2\mathbf{\hat{e}}_2+v_3\mathbf{\hat{e}}_3) \\ &=(u_2v_3 - u_3v_2)\mathbf{\hat{e}}_1+(u_3v_1 - u_1v_3)\mathbf{\hat{e}}_2+(u_1v_2 - u_2v_1)\mathbf{\hat{e}}_3 \end{align*} \]

Scalar Triple Product

In geometry, a parallelepiped is a three-dimensional figure formed by six parallelograms. The volume for a parallelepiped is: \[ V = \text{Base . Height} \]

Let a parallelepiped by given by three vectors $\vec u, \vec v, \vec w$ then we define the base: \[ \text{Base} = \text{Area of parallelogram with edges } \vec u, \vec v = \norm{ \mathbf{u} \times \mathbf{v}} \] and the height: \[ \text{Height} = \text{projection of } \vec w \text{ onto } \vec u \times \vec v=\norm{ \mathbf{w}} \left| \cos \theta \right| \] with $0\less\theta\less\pi$ the angle between $\vec w$ and $\vec u \times \vec v$. Putting these pieces together gives: \[ V = \text{Base . Height} = \norm{ \mathbf{u} \times \mathbf{v}} .\norm{ \mathbf{w}} \left|\cos \theta \right| = \left|(\vec u \times \vec v).\vec w\right| \]

Scalar Triple Product

\[ \begin{align*} (\vec u, \vec v, \vec w)=\vec u \cdot (\vec v \times \vec w) &= u_1(v_2w_3-v_3w_2)+u_2(v_3w_1-v_1w_3)+u_3(v_1w_2-v_2w_1) \\ &=\left| {\begin{array}{*{20}c} u_1 & u_2 & u_3 \\ v_1 & v_2 & v_3 \\ w_1 & w_2 & w_3 \\ \end{array} } \right| \end{align*} \] Note 1:each term $u_i(v_jw_k-v_kw_j)$ with $i,j,k$ all different and a permutation of indices such that an even number of interchanges of adjacent elements has a positive sign and odd number a negative sign
Note 2:$\times$ has a higher precedence than $\cdot$ so we may ommit the parentheses and write $\vec u \cdot \vec v \times \vec w$
Note 3:$\vec u \cdot (\vec v \times \vec w)=\vec w \cdot(\vec u \times \vec v)$
From the definition of determinants is follows that an exchange of two rows changes the sign of the determinant: \[ (\vec u, \vec v, \vec w)=(\vec v,\vec w,\vec u)=(\vec w, \vec u,\vec v) \text{ even number of exchanges } \] In other words, the scalar triple product does not change under cyclic permutation of the vectors or when $\cdot$ and $\times$ symbols are switched.

Triple Vector Product

The triple vector product is defined by \[ \vec u \times (\vec v \times \vec w) \]

Analyzing this product we notice that from the definition of the cross product the triple product is perpendicular to $(\vec v \times \vec w)$. All vectors perpendicular to this vector lie in the plane spanned by the two vectors $\vec v$ and $\vec w$. So, the triple product must be some linear combination of these vectors: \[ \vec u \times (\vec v \times \vec w) = p\vec v + q \vec w \] with $p$ and $q$ some numbers. Multiplying the triple product with $\vec u$ gives $0$ because both vectors are orthogonal to each other. This gives: \[ 0=p (\vec u \cdot \vec v) + q (\vec u \cdot \vec w) \] Let us take $p=\lambda (\vec u \cdot \vec w)$ and $q=-\lambda (\vec u \cdot \vec v)$ we then get for the triple product: \begin{equation} \vec u \times (\vec v \times \vec w)= \lambda\left[(\vec u \cdot \vec w)\vec v - (\vec u \cdot \vec v) \vec w\right] \label{eq03} \end{equation} It can be shown that $\lambda$ is independent of $\vec u,\vec v,\vec w$ and $\lambda=1$ we therefore have following result:

Triple Vector Product
\[ \vec u \times (\vec v \times \vec w)=(\vec u \cdot \vec w)\vec v - (\vec u \cdot \vec v) \vec w \] In words: the tripe vector product is the middle vector times the dot product of the other two minus the other vector between parentheses times the dot product of the other two vectors.
Proof

Higher dimensional cross product

It can be shown (see[2]) that the bilinear cross product , with the properties as defined ealier for 3 dimensions, only exists for $n=3$ or $n=7$. We will define a multilinear $(n-1)$ cross product for n-dimensional space. We have already established the following fact for the volume of a parallelepiped: \[ \text{vol}^2\;P_k=\det G(\vec v_1,\cdots,\vec v_k) \] We will use this for the definition of the magnitude of the cross product in n dimensions.

Definition: Cross Product - N dimensional

A cross product is defined for $n>1$ which assigns to any tuple of $(n-1)$ vectors $\vec v_1, \vec v_2, \cdots ,\vec v_{n-1} \in \mathbb{R}^n$ a vector $\vec v_1 \times \vec v_2 \times \cdots \times \vec v_{n-1} \in \mathbb{R}^n$ such that the following three properies hold:

  • $\vec v_1 \times \vec v_2 \times \cdots \times \vec v_{n-1}$ is a multilinear function
  • $(\vec v_1 \times \vec v_2 \times \cdots \times \vec v_{n-1})\cdot \vec v_i=0$, the cross product is perpendicular to $\vec v_1, \vec v_2, \cdots ,\vec v_{n-1}$
  • $\norm{\vec v_1 \times \vec v_2 \times \cdots \times \vec v_{n-1}}^2=\det G(\vec v_1,\cdots,\vec v_{n-1})$, the magnitude of the cross product is the volume of the parallelepiped spanned by the vectors $(\vec v_1,\cdots,\vec v_{n-1})$
  • $(\mathbf{\hat{e}_1} \times \mathbf{\hat{e}_2} \cdots \times \mathbf{\hat{e}_{n-1}})=\mathbf{\hat{e}_n}$ where $\{\mathbf{\hat{e}_1},\mathbf{\hat{e}_2} \cdots ,\mathbf{\hat{e}_{n}}\}$ is an orthonormal basis for $\mathbb{R}^n$

We see clearly the geometric nature of the cross product by its relation to the volume. But the volume relates to the algebraic notion of the determinant.

References



Copyright 2013 Jacq Krol. All rights reserved.